Question: Given that the point $(4,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=3f(4x)+5$. What is the sum of the coordinates of that point?
Answer: Since $(4,7)$ is on the graph of $y=f(x)$, we know  \[7=f(4).\]Using that $f(4\cdot1)=7$, we can also say \[3f(4\cdot1)+5=3\cdot7+5=26=2\cdot13.\]Therefore $(x,y)=(1,13)$ is on the graph of  \[2y=3f(4\cdot x)+5.\]The sum of these coordinates is $1+13=\boxed{14}$.